There are four possible operations of arity 1. What are they? I know negation is one but cant think of anything else? I need three more!
 A: Hint: If you mean truth functions, this is the same as the number of functions $\{ 0 , 1 \} \to \{ 0 , 1 \}$.  Forgetting about the "meaning" of these truth functions might make it easier.
A: 1) $f(0)=0,\ f(1)=0$
2) $f(0)=0,\ f(1)=1$
3) $f(0)=1,\ f(1)=0$ (negation)
4) $f(0)=1,\ f(1)=1$
A: The four possible unary operations are;
$T \mapsto T$
$F\mapsto   F$
$T \mapsto  F$
$F \mapsto  F$
$T \mapsto  T$
$F \mapsto  T$
$T \mapsto  F$
$F \mapsto  T$
A: $$f_i: \{T, F\} \to \{T, F\}$$
$$\text{_________________________}$$
$$f_1(T) = T, \quad f_1(F) = T\tag{T}$$
$$f_2(T) = T, \quad f_2(F) = F\tag{Identity}$$
$$f_3(T) = F, \quad f_3(F) = T\tag{Negation}$$
$$f_4(T) = F, \quad f_4(F) = F\tag{F}$$
$$\text{__________________________}$$
$$|\{f_i \mid f_i: \{T, F\} \to \{T, F\}\}| = 4$$
$f_1$ always returns the value of $T$: 
$f_2$ returns the truth value of the input.
$f_3$ returns the value of $\lnot$ (input)
$f_4$ always returns the value of $F$
A: If you go outside the Boolean operations
to the reals,
there are an unlimited number of operations:
increment, reciprocal, factorial, trig,
blah, blah.
As usual, it's a requirements problem
(too many years in aerospace).
